Concepts Of Physics MCQ Edition [Volume 2]

Concepts Of Physics MCQ Edition [Volume 2] · Physics · The Special Theory of Relativity

Determine the mass, the kinetic energy, and the momentum of an...

Determine the mass, the kinetic energy, and the momentum of an electron travelling at

0.8c

Options

A $9.1 \times 10^{-31} \text{ kg}$ , $2.6 \times 10^{-14} \text{ J}$ , $2.18 \times 10^{-22} \text{ kg m s}^{-1}$
B $15.2 \times 10^{-31} \text{ kg}$ , $13.6 \times 10^{-14} \text{ J}$ , $3.65 \times 10^{-22} \text{ kg m s}^{-1}$
C $15.2 \times 10^{-31} \text{ kg}$ , $4.4 \times 10^{-14} \text{ J}$ , $3.65 \times 10^{-22} \text{ kg m s}^{-1}$
D $15.2 \times 10^{-31} \text{ kg}$ , $5.5 \times 10^{-14} \text{ J}$ , $3.65 \times 10^{-22} \text{ kg m s}^{-1}$

D. $15.2 \times 10^{-31} \text{ kg}$ , $5.5 \times 10^{-14} \text{ J}$ , $3.65 \times 10^{-22} \text{ kg m s}^{-1}$

The rest mass of an electron is

m_0 = 9.1 \times 10^{-31} \text{ kg}

.

The relativistic factor

\gamma

is given by:

\gamma = \dfrac{1}{\sqrt{1 - \dfrac{v^2}{c^2}}} = \dfrac{1}{\sqrt{1 - (0.8)^2}} = \dfrac{1}{0.6} = \dfrac{5}{3}

The relativistic mass

m

is:

m = \gamma m_0 = \dfrac{5}{3} \times 9.1 \times 10^{-31} \text{ kg} \approx 15.2 \times 10^{-31} \text{ kg}

The kinetic energy

K

is:

K = (\gamma - 1)m_0 c^2 = \left(\dfrac{5}{3} - 1\right) \times 9.1 \times 10^{-31} \times (3 \times 10^8)^2

K = \dfrac{2}{3} \times 9.1 \times 10^{-31} \times 9 \times 10^{16} = 54.6 \times 10^{-15} \text{ J} \approx 5.5 \times 10^{-14} \text{ J}

The momentum

p

is:

p = mv = \left(\dfrac{5}{3} \times 9.1 \times 10^{-31}\right) \times (0.8 \times 3 \times 10^8)

p = 36.4 \times 10^{-23} \text{ kg m s}^{-1} \approx 3.65 \times 10^{-22} \text{ kg m s}^{-1}